LMFDB - Embedded newform 177.4.d.c.176.16

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.4433380710$
Analytic rank:	$0$
Dimension:	$52$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			176.16
Character	$\chi$	$=$	177.176
Dual form			177.4.d.c.176.15

$q$-expansion

$f(q)$	$=$	$q-2.68371 q^{2} +(-3.19645 + 4.09667i) q^{3} -0.797724 q^{4} +15.8157i q^{5} +(8.57832 - 10.9943i) q^{6} +25.4256 q^{7} +23.6105 q^{8} +(-6.56545 - 26.1896i) q^{9} +O(q^{10})$	\(q-2.68371 q^{2} +(-3.19645 + 4.09667i) q^{3} -0.797724 q^{4} +15.8157i q^{5} +(8.57832 - 10.9943i) q^{6} +25.4256 q^{7} +23.6105 q^{8} +(-6.56545 - 26.1896i) q^{9} -42.4446i q^{10} +64.0388 q^{11} +(2.54988 - 3.26801i) q^{12} +26.7287i q^{13} -68.2349 q^{14} +(-64.7916 - 50.5539i) q^{15} -56.9818 q^{16} +54.3878i q^{17} +(17.6197 + 70.2852i) q^{18} +47.4395 q^{19} -12.6165i q^{20} +(-81.2717 + 104.160i) q^{21} -171.861 q^{22} +174.401 q^{23} +(-75.4697 + 96.7245i) q^{24} -125.135 q^{25} -71.7320i q^{26} +(128.276 + 56.8171i) q^{27} -20.2826 q^{28} +169.829i q^{29} +(173.882 + 135.672i) q^{30} -217.458i q^{31} -35.9615 q^{32} +(-204.697 + 262.346i) q^{33} -145.961i q^{34} +402.123i q^{35} +(5.23742 + 20.8921i) q^{36} +122.447i q^{37} -127.314 q^{38} +(-109.499 - 85.4369i) q^{39} +373.416i q^{40} -368.046i q^{41} +(218.109 - 279.536i) q^{42} -3.01728i q^{43} -51.0853 q^{44} +(414.206 - 103.837i) q^{45} -468.041 q^{46} -314.192 q^{47} +(182.139 - 233.436i) q^{48} +303.463 q^{49} +335.826 q^{50} +(-222.809 - 173.848i) q^{51} -21.3221i q^{52} +468.533i q^{53} +(-344.256 - 152.480i) q^{54} +1012.82i q^{55} +600.312 q^{56} +(-151.638 + 194.344i) q^{57} -455.771i q^{58} +(-224.948 + 393.417i) q^{59} +(51.6858 + 40.3281i) q^{60} -558.899i q^{61} +583.593i q^{62} +(-166.931 - 665.887i) q^{63} +552.365 q^{64} -422.733 q^{65} +(549.346 - 704.060i) q^{66} -284.086i q^{67} -43.3865i q^{68} +(-557.464 + 714.464i) q^{69} -1079.18i q^{70} -279.692i q^{71} +(-155.014 - 618.349i) q^{72} -877.906i q^{73} -328.611i q^{74} +(399.988 - 512.638i) q^{75} -37.8436 q^{76} +1628.23 q^{77} +(293.863 + 229.288i) q^{78} -63.0858 q^{79} -901.206i q^{80} +(-642.790 + 343.893i) q^{81} +987.728i q^{82} +941.269 q^{83} +(64.8323 - 83.0913i) q^{84} -860.180 q^{85} +8.09748i q^{86} +(-695.733 - 542.849i) q^{87} +1511.99 q^{88} -199.336 q^{89} +(-1111.61 + 278.668i) q^{90} +679.594i q^{91} -139.124 q^{92} +(890.854 + 695.093i) q^{93} +843.199 q^{94} +750.288i q^{95} +(114.949 - 147.322i) q^{96} +432.568i q^{97} -814.404 q^{98} +(-420.444 - 1677.15i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + O(q^{10}) $	$ 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + 28q^{12} + 114q^{15} + 484q^{16} - 184q^{19} - 758q^{21} - 60q^{22} + 36q^{25} + 742q^{27} - 4q^{28} - 888q^{36} + 1402q^{45} - 660q^{46} - 488q^{48} - 924q^{49} - 1772q^{51} - 630q^{57} - 1880q^{60} - 212q^{63} + 7648q^{64} + 1316q^{66} - 1556q^{75} - 5680q^{76} + 3224q^{78} - 1504q^{79} - 276q^{81} + 1228q^{84} - 848q^{85} + 3598q^{87} + 5760q^{88} + 888q^{94} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−2.68371			−0.948833			−0.474417	−	0.880300i	$-0.657341\pi$
$2$	−2.68371			−0.948833			−0.474417	+	0.880300i	$0.657341\pi$
$3$	−3.19645	+	4.09667i	−0.615157	+	0.788405i
$3$	−3.19645	+	4.09667i	−0.615157	+	0.788405i
$4$	−0.797724			−0.0997154
$5$			15.8157i			1.41460i	0.706915	+	0.707298i	$0.250086\pi$
$5$			15.8157i			1.41460i	−0.706915	+	0.707298i	$0.749914\pi$
$6$	8.57832	−	10.9943i	0.583681	−	0.748065i
$7$	25.4256			1.37285			0.686427	−	0.727198i	$-0.259178\pi$
$7$	25.4256			1.37285			0.686427	+	0.727198i	$0.259178\pi$
$8$	23.6105			1.04345
$9$	−6.56545	−	26.1896i	−0.243165	−	0.969985i
$10$		−	42.4446i		−	1.34222i
$11$	64.0388			1.75531			0.877656	−	0.479291i	$-0.159106\pi$
$11$	64.0388			1.75531			0.877656	+	0.479291i	$0.159106\pi$
$12$	2.54988	−	3.26801i	0.0613406	−	0.0786162i
$13$			26.7287i			0.570247i	0.958491	+	0.285124i	$0.0920346\pi$
$13$			26.7287i			0.570247i	−0.958491	+	0.285124i	$0.907965\pi$
$14$	−68.2349			−1.30261
$15$	−64.7916	−	50.5539i	−1.11527	−	0.870198i
$16$	−56.9818			−0.890341
$17$			54.3878i			0.775940i	0.921672	+	0.387970i	$0.126824\pi$
$17$			54.3878i			0.775940i	−0.921672	+	0.387970i	$0.873176\pi$
$18$	17.6197	+	70.2852i	0.230723	+	0.920354i
$19$	47.4395			0.572809			0.286405	−	0.958109i	$-0.407540\pi$
$19$	47.4395			0.572809			0.286405	+	0.958109i	$0.407540\pi$
$20$		−	12.6165i		−	0.141057i
$21$	−81.2717	+	104.160i	−0.844521	+	1.08237i
$22$	−171.861			−1.66550
$23$	174.401			1.58109			0.790546	−	0.612402i	$-0.209797\pi$
$23$	174.401			1.58109			0.790546	+	0.612402i	$0.209797\pi$
$24$	−75.4697	+	96.7245i	−0.641883	+	0.822659i
$25$	−125.135			−1.00108
$26$		−	71.7320i		−	0.541069i
$27$	128.276	+	56.8171i	0.914325	+	0.404980i
$28$	−20.2826			−0.136895
$29$			169.829i			1.08746i	0.839259	+	0.543732i	$0.182989\pi$
$29$			169.829i			1.08746i	−0.839259	+	0.543732i	$0.817011\pi$
$30$	173.882	+	135.672i	1.05821	+	0.825673i
$31$		−	217.458i		−	1.25989i	−0.776639	−	0.629945i	$-0.783077\pi$
$31$		−	217.458i		−	1.25989i	0.776639	−	0.629945i	$-0.216923\pi$
$32$	−35.9615			−0.198661
$33$	−204.697	+	262.346i	−1.07979	+	1.38390i
$34$		−	145.961i		−	0.736238i
$35$			402.123i			1.94204i
$36$	5.23742	+	20.8921i	0.0242473	+	0.0967225i
$37$			122.447i			0.544057i	0.962289	+	0.272028i	$0.0876944\pi$
$37$			122.447i			0.544057i	−0.962289	+	0.272028i	$0.912306\pi$
$38$	−127.314			−0.543500
$39$	−109.499	−	85.4369i	−0.449586	−	0.350791i
$40$			373.416i			1.47606i
$41$		−	368.046i		−	1.40193i	−0.713195	−	0.700966i	$-0.752753\pi$
$41$		−	368.046i		−	1.40193i	0.713195	−	0.700966i	$-0.247247\pi$
$42$	218.109	−	279.536i	0.801309	−	1.02698i
$43$		−	3.01728i		−	0.0107007i	−0.999986	−	0.00535035i	$-0.998297\pi$
$43$		−	3.01728i		−	0.0107007i	0.999986	−	0.00535035i	$-0.00170308\pi$
$44$	−51.0853			−0.175032
$45$	414.206	−	103.837i	1.37214	−	0.343980i
$46$	−468.041			−1.50019
$47$	−314.192			−0.975098			−0.487549	−	0.873096i	$-0.662109\pi$
$47$	−314.192			−0.975098			−0.487549	+	0.873096i	$0.662109\pi$
$48$	182.139	−	233.436i	0.547699	−	0.701950i
$49$	303.463			0.884731
$50$	335.826			0.949860
$51$	−222.809	−	173.848i	−0.611755	−	0.477325i
$52$		−	21.3221i		−	0.0568624i
$53$			468.533i			1.21430i	0.794587	+	0.607150i	$0.207687\pi$
$53$			468.533i			1.21430i	−0.794587	+	0.607150i	$0.792313\pi$
$54$	−344.256	−	152.480i	−0.867542	−	0.384259i
$55$			1012.82i			2.48306i
$56$	600.312			1.43250
$57$	−151.638	+	194.344i	−0.352367	+	0.451606i
$58$		−	455.771i		−	1.03182i
$59$	−224.948	+	393.417i	−0.496369	+	0.868112i
$59$	−224.948	+	393.417i	−0.496369	+	0.868112i
$60$	51.6858	+	40.3281i	0.111210	+	0.0867722i
$61$		−	558.899i		−	1.17311i	−0.809910	−	0.586554i	$-0.800484\pi$
$61$		−	558.899i		−	1.17311i	0.809910	−	0.586554i	$-0.199516\pi$
$62$			583.593i			1.19543i
$63$	−166.931	−	665.887i	−0.333830	−	1.33165i
$64$	552.365			1.07884
$65$	−422.733			−0.806669
$66$	549.346	−	704.060i	1.02454	−	1.31309i
$67$		−	284.086i		−	0.518009i	−0.965876	−	0.259004i	$-0.916606\pi$
$67$		−	284.086i		−	0.518009i	0.965876	−	0.259004i	$-0.0833944\pi$
$68$		−	43.3865i		−	0.0773732i
$69$	−557.464	+	714.464i	−0.972619	+	1.24654i
$70$		−	1079.18i		−	1.84267i
$71$		−	279.692i		−	0.467512i	−0.972295	−	0.233756i	$-0.924898\pi$
$71$		−	279.692i		−	0.467512i	0.972295	−	0.233756i	$-0.0751017\pi$
$72$	−155.014	−	618.349i	−0.253730	−	1.01213i
$73$		−	877.906i		−	1.40755i	−0.710423	−	0.703775i	$-0.751496\pi$
$73$		−	877.906i		−	1.40755i	0.710423	−	0.703775i	$-0.248504\pi$
$74$		−	328.611i		−	0.516219i
$75$	399.988	−	512.638i	0.615822	−	0.789258i
$76$	−37.8436			−0.0571179
$77$	1628.23			2.40979
$78$	293.863	+	229.288i	0.426582	+	0.332842i
$79$	−63.0858			−0.0898445			−0.0449222	−	0.998990i	$-0.514304\pi$
$79$	−63.0858			−0.0898445			−0.0449222	+	0.998990i	$0.514304\pi$
$80$		−	901.206i		−	1.25947i
$81$	−642.790	+	343.893i	−0.881742	+	0.471733i
$82$			987.728i			1.33020i
$83$	941.269			1.24479			0.622396	−	0.782703i	$-0.286159\pi$
$83$	941.269			1.24479			0.622396	+	0.782703i	$0.286159\pi$
$84$	64.8323	−	83.0913i	0.0842118	−	0.107929i
$85$	−860.180			−1.09764
$86$			8.09748i			0.0101532i
$87$	−695.733	−	542.849i	−0.857361	−	0.668960i
$88$	1511.99			1.83157
$89$	−199.336			−0.237411			−0.118706	−	0.992930i	$-0.537874\pi$
$89$	−199.336			−0.237411			−0.118706	+	0.992930i	$0.537874\pi$
$90$	−1111.61	+	278.668i	−1.30193	+	0.326380i
$91$			679.594i			0.782867i
$92$	−139.124			−0.157659
$93$	890.854	+	695.093i	0.993304	+	0.775030i
$94$	843.199			0.925206
$95$			750.288i			0.810294i
$96$	114.949	−	147.322i	0.122208	−	0.156625i
$97$			432.568i			0.452790i	0.974036	+	0.226395i	$0.0726939\pi$
$97$			432.568i			0.452790i	−0.974036	+	0.226395i	$0.927306\pi$
$98$	−814.404			−0.839462
$99$	−420.444	−	1677.15i	−0.426830	−	1.70263i
$100$	99.8234			0.0998234
$101$	−1768.19			−1.74200			−0.870999	−	0.491285i	$-0.836528\pi$
$101$	−1768.19			−1.74200			−0.870999	+	0.491285i	$0.836528\pi$
$102$	597.954	+	466.556i	0.580454	+	0.452902i
$103$		−	947.121i		−	0.906044i	−0.891499	−	0.453022i	$-0.850346\pi$
$103$		−	947.121i		−	0.906044i	0.891499	−	0.453022i	$-0.149654\pi$
$104$			631.078i			0.595022i
$105$	−1647.37	−	1285.37i	−1.53111	−	1.19466i
$106$		−	1257.40i		−	1.15217i
$107$			1078.46i			0.974379i	0.873296	+	0.487190i	$0.161978\pi$
$107$			1078.46i			0.974379i	−0.873296	+	0.487190i	$0.838022\pi$
$108$	−102.329	−	45.3244i	−0.0911724	−	0.0403828i
$109$			1042.51i			0.916096i	0.888927	+	0.458048i	$0.151451\pi$
$109$			1042.51i			0.916096i	−0.888927	+	0.458048i	$0.848549\pi$
$110$		−	2718.10i		−	2.35601i
$111$	−501.624	−	391.394i	−0.428937	−	0.334680i
$112$	−1448.80			−1.22231
$113$	733.402			0.610555			0.305277	−	0.952263i	$-0.401251\pi$
$113$	733.402			0.610555			0.305277	+	0.952263i	$0.401251\pi$
$114$	406.952	−	521.563i	0.334338	−	0.428498i
$115$			2758.27i			2.23661i
$116$		−	135.476i		−	0.108437i
$117$	700.014	−	175.486i	0.553131	−	0.138664i
$118$	603.695	−	1055.82i	0.470972	−	0.823693i
$119$			1382.84i			1.06525i
$120$	−1529.76	−	1193.60i	−1.16373	−	0.908005i
$121$	2769.97			2.08112
$122$			1499.92i			1.11308i
$123$	1507.77	+	1176.44i	1.10529	+	0.862407i
$124$			173.471i			0.125631i
$125$		−	2.14016i		−	0.00153137i
$126$	447.993	+	1787.04i	0.316749	+	1.26351i
$127$	−365.016			−0.255039			−0.127520	−	0.991836i	$-0.540702\pi$
$127$	−365.016			−0.255039			−0.127520	+	0.991836i	$0.540702\pi$
$128$	−1194.69			−0.824976
$129$	12.3608	+	9.64457i	0.00843649	+	0.00658261i
$130$	1134.49			0.765395
$131$	263.810			0.175948			0.0879741	−	0.996123i	$-0.471961\pi$
$131$	263.810			0.175948			0.0879741	+	0.996123i	$0.471961\pi$
$132$	163.291	−	209.280i	0.107672	−	0.137996i
$133$	1206.18			0.786384
$134$			762.403i			0.491504i
$135$	−898.601	+	2028.78i	−0.572883	+	1.29340i
$136$			1284.12i			0.809652i
$137$		−	1218.37i		−	0.759799i	−0.925028	−	0.379900i	$-0.875959\pi$
$137$		−	1218.37i		−	0.759799i	0.925028	−	0.379900i	$-0.124041\pi$
$138$	1496.07	−	1917.41i	0.922854	−	1.18276i
$139$	−1137.74			−0.694256			−0.347128	−	0.937818i	$-0.612843\pi$
$139$	−1137.74			−0.694256			−0.347128	+	0.937818i	$0.612843\pi$
$140$		−	320.783i		−	0.193651i
$141$	1004.30	−	1287.14i	0.599838	−	0.768772i
$142$			750.612i			0.443591i
$143$			1711.68i			1.00096i
$144$	374.112	+	1492.33i	0.216500	+	0.863618i
$145$	−2685.96			−1.53832
$146$			2356.04i			1.33553i
$147$	−970.002	+	1243.19i	−0.544248	+	0.697526i
$148$		−	97.6785i		−	0.0542509i
$149$	−991.213			−0.544989			−0.272494	−	0.962157i	$-0.587849\pi$
$149$	−991.213			−0.544989			−0.272494	+	0.962157i	$0.587849\pi$
$150$	−1073.45	+	1375.77i	−0.584313	+	0.748875i
$151$			3527.16i			1.90090i	0.310874	+	0.950451i	$0.399378\pi$
$151$			3527.16i			1.90090i	−0.310874	+	0.950451i	$0.600622\pi$
$152$	1120.07			0.597696
$153$	1424.40	−	357.081i	0.752650	−	0.188681i
$154$	−4369.68			−2.28649
$155$	3439.24			1.78224
$156$	87.3498	+	68.1551i	0.0448306	+	0.0349793i
$157$		−	1442.44i		−	0.733241i	−0.930371	−	0.366621i	$-0.880515\pi$
$157$		−	1442.44i		−	0.733241i	0.930371	−	0.366621i	$-0.119485\pi$
$158$	169.304			0.0852474
$159$	−1919.43	−	1497.64i	−0.957361	−	0.746985i
$160$		−	568.755i		−	0.281025i
$161$	4434.26			2.17061
$162$	1725.06	−	922.908i	0.836626	−	0.447596i
$163$	−2181.89			−1.04846			−0.524229	−	0.851577i	$-0.675646\pi$
$163$	−2181.89			−1.04846			−0.524229	+	0.851577i	$0.675646\pi$
$164$			293.599i			0.139794i
$165$	−4149.18	−	3237.41i	−1.95766	−	1.52747i
$166$	−2526.09			−1.18110
$167$		−	2121.08i		−	0.982840i	−0.870923	−	0.491420i	$-0.836478\pi$
$167$		−	2121.08i		−	0.982840i	0.870923	−	0.491420i	$-0.163522\pi$
$168$	−1918.86	+	2459.28i	−0.881212	+	1.12939i
$169$	1482.58			0.674818
$170$	2308.47			1.04148
$171$	−311.462	−	1242.42i	−0.139287	−	0.555616i
$172$			2.40695i			0.00106703i
$173$	−3125.58			−1.37360			−0.686801	−	0.726845i	$-0.740986\pi$
$173$	−3125.58			−1.37360			−0.686801	+	0.726845i	$0.740986\pi$
$174$	1867.14	+	1456.85i	0.813493	+	0.634732i
$175$	−3181.64			−1.37434
$176$	−3649.05			−1.56283
$177$	−892.667	−	2179.08i	−0.379079	−	0.925364i
$178$	534.959			0.225263
$179$	−509.207			−0.212625			−0.106313	−	0.994333i	$-0.533904\pi$
$179$	−509.207			−0.212625			−0.106313	+	0.994333i	$0.533904\pi$
$180$	−330.422	+	82.8332i	−0.136823	+	0.0343001i
$181$	454.608			0.186689			0.0933445	−	0.995634i	$-0.470244\pi$
$181$	454.608			0.186689			0.0933445	+	0.995634i	$0.470244\pi$
$182$		−	1823.83i		−	0.742810i
$183$	2289.62	+	1786.49i	0.924885	+	0.721645i
$184$	4117.70			1.64979
$185$	−1936.57			−0.769621
$186$	−2390.79	−	1865.43i	−0.942480	−	0.735374i
$187$			3482.93i			1.36202i
$188$	250.638			0.0972324
$189$	3261.51	+	1444.61i	1.25524	+	0.555979i
$190$		−	2013.55i		−	0.768833i
$191$	162.666			0.0616234			0.0308117	−	0.999525i	$-0.490191\pi$
$191$	162.666			0.0616234			0.0308117	+	0.999525i	$0.490191\pi$
$192$	−1765.61	+	2262.86i	−0.663654	+	0.850561i
$193$	−2139.02			−0.797773			−0.398887	−	0.917000i	$-0.630603\pi$
$193$	−2139.02			−0.797773			−0.398887	+	0.917000i	$0.630603\pi$
$194$		−	1160.88i		−	0.429622i
$195$	1351.24	−	1731.80i	0.496228	−	0.635982i
$196$	−242.079			−0.0882213
$197$		−	5343.19i		−	1.93242i	−0.257755	−	0.966210i	$-0.582983\pi$
$197$		−	5343.19i		−	1.93242i	0.257755	−	0.966210i	$-0.417017\pi$
$198$	1128.35	+	4500.98i	0.404991	+	1.61551i
$199$	−3654.95			−1.30197			−0.650987	−	0.759089i	$-0.725645\pi$
$199$	−3654.95			−1.30197			−0.650987	+	0.759089i	$0.725645\pi$
$200$	−2954.51			−1.04458
$201$	1163.81	+	908.065i	0.408401	+	0.318657i
$202$	4745.31			1.65287
$203$			4318.01i			1.49293i
$204$	177.740	+	138.682i	0.0610014	+	0.0475967i
$205$	5820.90			1.98317
$206$			2541.79i			0.859685i
$207$	−1145.02	−	4567.49i	−0.384466	−	1.53364i
$208$		−	1523.05i		−	0.507715i
$209$	3037.97			1.00546
$210$	4421.05	+	3449.54i	1.45277	+	1.13353i
$211$		−	3655.96i		−	1.19283i	−0.802677	−	0.596415i	$-0.796591\pi$
$211$		−	3655.96i		−	1.19283i	0.802677	−	0.596415i	$-0.203409\pi$
$212$		−	373.760i		−	0.121085i
$213$	1145.81	+	894.022i	0.368589	+	0.287593i
$214$		−	2894.27i		−	0.924524i
$215$	47.7202			0.0151372
$216$	3028.67	+	1341.48i	0.954050	+	0.422575i
$217$		−	5529.01i		−	1.72965i
$218$		−	2797.79i		−	0.869222i
$219$	3596.49	+	2806.18i	1.10972	+	0.865864i
$220$		−	807.948i		−	0.247599i
$221$	−1453.72			−0.442478
$222$	1346.21	+	1050.39i	0.406990	+	0.317556i
$223$	111.332			0.0334319			0.0167160	−	0.999860i	$-0.494679\pi$
$223$	111.332			0.0334319			0.0167160	+	0.999860i	$0.494679\pi$
$224$	−914.344			−0.272733
$225$	821.570	+	3277.24i	0.243428	+	0.971035i
$226$	−1968.24			−0.579315
$227$	1122.58			0.328229			0.164115	−	0.986441i	$-0.447523\pi$
$227$	1122.58			0.328229			0.164115	+	0.986441i	$0.447523\pi$
$228$	120.965	−	155.033i	0.0351365	−	0.0450320i
$229$		−	113.487i		−	0.0327485i	−0.999866	−	0.0163743i	$-0.994788\pi$
$229$		−	113.487i		−	0.0327485i	0.999866	−	0.0163743i	$-0.00521232\pi$
$230$		−	7402.38i		−	2.12217i
$231$	−5204.54	+	6670.31i	−1.48240	+	1.89989i
$232$			4009.74i			1.13471i
$233$	704.563			0.198101			0.0990503	−	0.995082i	$-0.468420\pi$
$233$	704.563			0.198101			0.0990503	+	0.995082i	$0.468420\pi$
$234$	−1878.63	+	470.953i	−0.524829	+	0.131569i
$235$		−	4969.16i		−	1.37937i
$236$	179.447	−	313.838i	0.0494957	−	0.0865641i
$237$	201.651	−	258.442i	0.0552684	−	0.0708338i
$238$		−	3711.15i		−	1.01075i
$239$		−	5082.40i		−	1.37554i	−0.725930	−	0.687768i	$-0.758591\pi$
$239$		−	5082.40i		−	1.37554i	0.725930	−	0.687768i	$-0.241409\pi$
$240$	3691.95	+	2880.66i	0.992975	+	0.774773i
$241$	3832.06			1.02425			0.512126	−	0.858910i	$-0.328858\pi$
$241$	3832.06			1.02425			0.512126	+	0.858910i	$0.328858\pi$
$242$	−7433.79			−1.97464
$243$	645.826	−	3732.53i	0.170493	−	0.985359i
$244$			445.847i			0.116977i
$245$			4799.46i			1.25154i
$246$	−4046.40	−	3157.22i	−1.04874	−	0.818281i
$247$			1268.00i			0.326643i
$248$		−	5134.29i		−	1.31463i
$249$	−3008.72	+	3856.07i	−0.765742	+	0.981400i
$250$			5.74355i			0.00145302i
$251$			2141.56i			0.538542i	0.963064	+	0.269271i	$0.0867828\pi$
$251$			2141.56i			0.538542i	−0.963064	+	0.269271i	$0.913217\pi$
$252$	133.165	+	531.194i	0.0332880	+	0.132786i
$253$	11168.4			2.77531
$254$	979.596			0.241990
$255$	2749.52	−	3523.87i	0.675222	−	0.865387i
$256$	−1212.72			−0.296073
$257$			2229.18i			0.541059i	0.962712	+	0.270530i	$0.0871988\pi$
$257$			2229.18i			0.541059i	−0.962712	+	0.270530i	$0.912801\pi$
$258$	−33.1727	−	25.8832i	−0.00800482	−	0.00624580i
$259$			3113.28i			0.746911i
$260$	337.224			0.0804374
$261$	4447.75	−	1115.00i	1.05482	−	0.264433i
$262$	−707.989			−0.166945
$263$		−	2699.89i		−	0.633013i	−0.948590	−	0.316506i	$-0.897490\pi$
$263$		−	2699.89i		−	0.633013i	0.948590	−	0.316506i	$-0.102510\pi$
$264$	−4832.99	+	6194.12i	−1.12670	+	1.44402i
$265$	−7410.16			−1.71775
$266$	−3237.03			−0.746147
$267$	637.167	−	816.615i	0.146045	−	0.187176i
$268$			226.622i			0.0516535i
$269$	−6427.83			−1.45692			−0.728460	−	0.685088i	$-0.759764\pi$
$269$	−6427.83			−1.45692			−0.728460	+	0.685088i	$0.759764\pi$
$270$	2411.58	−	5444.64i	0.543571	−	1.22722i
$271$	−2588.09			−0.580130			−0.290065	−	0.957007i	$-0.593677\pi$
$271$	−2588.09			−0.580130			−0.290065	+	0.957007i	$0.593677\pi$
$272$		−	3099.12i		−	0.690852i
$273$	−2784.08	−	2172.29i	−0.617216	−	0.481585i
$274$			3269.75i			0.720923i
$275$	−8013.52			−1.75721
$276$	444.702	−	569.945i	0.0969852	−	0.124299i
$277$	7307.96			1.58517			0.792586	−	0.609760i	$-0.208734\pi$
$277$	7307.96			1.58517			0.792586	+	0.609760i	$0.208734\pi$
$278$	3053.35			0.658733
$279$	−5695.14	+	1427.71i	−1.22208	+	0.306361i
$280$			9494.33i			2.02641i
$281$			4020.21i			0.853472i	0.904376	+	0.426736i	$0.140337\pi$
$281$			4020.21i			0.853472i	−0.904376	+	0.426736i	$0.859663\pi$
$282$	−2695.24	+	3454.31i	−0.569146	+	0.729437i
$283$		−	246.102i		−	0.0516934i	−0.999666	−	0.0258467i	$-0.991772\pi$
$283$		−	246.102i		−	0.0516934i	0.999666	−	0.0258467i	$-0.00822817\pi$
$284$			223.117i			0.0466182i
$285$	−3073.68	−	2398.25i	−0.638839	−	0.498457i
$286$		−	4593.63i		−	0.949746i
$287$		−	9357.81i		−	1.92465i
$288$	236.103	+	941.817i	0.0483074	+	0.192698i
$289$	1954.96			0.397917
$290$	7208.32			1.45961
$291$	−1772.09	−	1382.68i	−0.356982	−	0.278536i
$292$			700.326i			0.140354i
$293$		−	3404.05i		−	0.678726i	−0.940656	−	0.339363i	$-0.889789\pi$
$293$		−	3404.05i		−	0.678726i	0.940656	−	0.339363i	$-0.110211\pi$
$294$	2603.20	−	3336.35i	0.516400	−	0.661836i
$295$	−6222.16	−	3557.71i	−1.22803	−	0.702162i
$296$			2891.03i			0.567694i
$297$	8214.66	+	3638.50i	1.60493	+	0.710866i
$298$	2660.12			0.517103
$299$			4661.52i			0.901614i
$300$	−319.080	+	408.944i	−0.0614070	+	0.0787013i
$301$		−	76.7162i		−	0.0146905i
$302$		−	9465.86i		−	1.80364i
$303$	5651.94	−	7243.71i	1.07160	−	1.37340i
$304$	−2703.19			−0.509996
$305$	8839.35			1.65948
$306$	−3822.66	+	958.300i	−0.714140	+	0.179027i
$307$	4099.88			0.762190			0.381095	−	0.924536i	$-0.375547\pi$
$307$	4099.88			0.762190			0.381095	+	0.924536i	$0.375547\pi$
$308$	−1298.88			−0.240293
$309$	3880.04	+	3027.42i	0.714330	+	0.557359i
$310$	−9229.92			−1.69105
$311$			1445.91i			0.263633i	0.991274	+	0.131817i	$0.0420810\pi$
$311$			1445.91i			0.263633i	−0.991274	+	0.131817i	$0.957919\pi$
$312$	−2585.32	−	2017.21i	−0.469119	−	0.366032i
$313$			1949.24i			0.352005i	0.984390	+	0.176002i	$0.0563166\pi$
$313$			1949.24i			0.352005i	−0.984390	+	0.176002i	$0.943683\pi$
$314$			3871.07i			0.695724i
$315$	10531.4	−	2640.12i	1.88375	−	0.472235i
$316$	50.3251			0.00895888
$317$			8787.37i			1.55693i	0.627686	+	0.778466i	$0.284002\pi$
$317$			8787.37i			1.55693i	−0.627686	+	0.778466i	$0.715998\pi$
$318$	5151.17	+	4019.23i	0.908376	+	0.708764i
$319$			10875.6i			1.90884i
$320$			8736.02i			1.52612i
$321$	−4418.09	−	3447.24i	−0.768206	−	0.599396i
$322$	−11900.2			−2.05955
$323$			2580.13i			0.444466i
$324$	512.768	−	274.332i	0.0879233	−	0.0470390i
$325$		−	3344.71i		−	0.570864i
$326$	5855.54			0.994811
$327$	−4270.83	−	3332.33i	−0.722255	−	0.563542i
$328$		−	8689.76i		−	1.46284i
$329$	−7988.53			−1.33867
$330$	11135.2	+	8688.27i	1.85749	+	1.44931i
$331$	−2972.50			−0.493606			−0.246803	−	0.969066i	$-0.579380\pi$
$331$	−2972.50			−0.493606			−0.246803	+	0.969066i	$0.579380\pi$
$332$	−750.873			−0.124125
$333$	3206.83	−	803.917i	0.527727	−	0.132295i
$334$			5692.36i			0.932552i
$335$	4493.01			0.732773
$336$	4631.01	−	5935.26i	0.751912	−	0.963675i
$337$		−	4402.05i		−	0.711558i	−0.934570	−	0.355779i	$-0.884216\pi$
$337$		−	4402.05i		−	0.711558i	0.934570	−	0.355779i	$-0.115784\pi$
$338$	−3978.80			−0.640290
$339$	−2344.28	+	3004.51i	−0.375587	+	0.481364i
$340$	686.186			0.109452
$341$		−	13925.8i		−	2.21150i
$342$	835.872	+	3334.29i	0.132160	+	0.527187i
$343$	−1005.26			−0.158248
$344$		−	71.2394i		−	0.0111656i
$345$	−11299.7	−	8816.66i	−1.76335	−	1.37586i
$346$	8388.13			1.30332
$347$	4806.25			0.743554			0.371777	−	0.928322i	$-0.378749\pi$
$347$	4806.25			0.743554			0.371777	+	0.928322i	$0.378749\pi$
$348$	555.003	+	433.043i	0.0854922	+	0.0667056i
$349$		−	9011.33i		−	1.38214i	−0.722790	−	0.691068i	$-0.757141\pi$
$349$		−	9011.33i		−	1.38214i	0.722790	−	0.691068i	$-0.242859\pi$
$350$	8538.60			1.30402
$351$	−1518.65	+	3428.66i	−0.230939	+	0.521391i
$352$	−2302.93			−0.348712
$353$	−11298.7			−1.70359			−0.851797	−	0.523872i	$-0.824487\pi$
$353$	−11298.7			−1.70359			−0.851797	+	0.523872i	$0.824487\pi$
$354$	2395.66	+	5848.00i	0.359683	+	0.878017i
$355$	4423.52			0.661341
$356$	159.015			0.0236735
$357$	−5665.06	−	4420.19i	−0.839851	−	0.655298i
$358$	1366.56			0.201746
$359$			7880.94i			1.15861i	0.815112	+	0.579304i	$0.196676\pi$
$359$			7880.94i			1.15861i	−0.815112	+	0.579304i	$0.803324\pi$
$360$	9779.61	−	2451.64i	1.43175	−	0.358925i
$361$	−4608.49			−0.671890
$362$	−1220.03			−0.177137
$363$	−8854.06	+	11347.7i	−1.28021	+	1.64077i
$364$		−	542.129i		−	0.0780639i
$365$	13884.7			1.99111
$366$	−6144.68	−	4794.41i	−0.877561	−	0.684721i
$367$		−	4266.06i		−	0.606775i	−0.952867	−	0.303388i	$-0.901882\pi$
$367$		−	4266.06i		−	0.606775i	0.952867	−	0.303388i	$-0.0981177\pi$
$368$	−9937.69			−1.40771
$369$	−9638.99	+	2416.39i	−1.35985	+	0.340901i
$370$	5197.20			0.730242
$371$			11912.7i			1.66706i
$372$	−710.656	−	554.492i	−0.0990478	−	0.0772825i
$373$	5386.56			0.747736			0.373868	−	0.927482i	$-0.378031\pi$
$373$	5386.56			0.747736			0.373868	+	0.927482i	$0.378031\pi$
$374$		−	9347.17i		−	1.29233i
$375$	8.76751	+	6.84089i	0.00120734	+	0.000942032i
$376$	−7418.23			−1.01746
$377$	−4539.31			−0.620123
$378$	−8752.92	−	3876.91i	−1.19101	−	0.527531i
$379$	−2251.27			−0.305119			−0.152560	−	0.988294i	$-0.548752\pi$
$379$	−2251.27			−0.305119			−0.152560	+	0.988294i	$0.548752\pi$
$380$		−	598.522i		−	0.0807988i
$381$	1166.76	−	1495.35i	0.156889	−	0.201074i
$382$	−436.547			−0.0584703
$383$		−	7169.18i		−	0.956470i	−0.878232	−	0.478235i	$-0.841277\pi$
$383$		−	7169.18i		−	0.956470i	0.878232	−	0.478235i	$-0.158723\pi$
$384$	3818.77	−	4894.27i	0.507489	−	0.650415i
$385$			25751.5i			3.40888i
$386$	5740.51			0.756954
$387$	−79.0213	+	19.8098i	−0.0103795	+	0.00260204i
$388$		−	345.069i		−	0.0451501i
$389$		−	2424.33i		−	0.315986i	−0.987440	−	0.157993i	$-0.949498\pi$
$389$		−	2424.33i		−	0.315986i	0.987440	−	0.157993i	$-0.0505024\pi$
$390$	−3626.34	+	4647.63i	−0.470838	+	0.603441i
$391$			9485.29i			1.22683i
$392$	7164.90			0.923169
$393$	−843.255	+	1080.74i	−0.108236	+	0.138718i
$394$			14339.6i			1.83354i
$395$		−	997.745i		−	0.127094i
$396$	335.398	+	1337.90i	0.0425616	+	0.169778i
$397$		−	14687.4i		−	1.85677i	−0.371615	−	0.928387i	$-0.621196\pi$
$397$		−	14687.4i		−	1.85677i	0.371615	−	0.928387i	$-0.378804\pi$
$398$	9808.82			1.23536
$399$	−3855.49	+	4941.32i	−0.483749	+	0.619989i
$400$	7130.44			0.891305
$401$	9413.70			1.17231			0.586157	−	0.810198i	$-0.300640\pi$
$401$	9413.70			1.17231			0.586157	+	0.810198i	$0.300640\pi$
$402$	−3123.31	−	2436.98i	−0.387504	−	0.302352i
$403$	5812.37			0.718449
$404$	1410.53			0.173704
$405$	−5438.90	−	10166.1i	−0.667311	−	1.24731i
$406$		−	11588.3i		−	1.41654i
$407$			7841.34i			0.954989i
$408$	−5260.64	−	4104.63i	−0.638334	−	0.498063i
$409$			7052.63i			0.852641i	0.904572	+	0.426321i	$0.140190\pi$
$409$			7052.63i			0.852641i	−0.904572	+	0.426321i	$0.859810\pi$
$410$	−15621.6			−1.88169
$411$	4991.27	+	3894.46i	0.599029	+	0.467395i
$412$			755.540i			0.0903466i
$413$	−5719.45	+	10002.9i	−0.681443	+	1.19179i
$414$	3072.90	+	12257.8i	0.364794	+	1.45516i
$415$			14886.8i			1.76088i
$416$		−	961.205i		−	0.113286i
$417$	3636.72	−	4660.94i	0.427076	−	0.547355i
$418$	−8153.02			−0.954012
$419$	8388.86			0.978097			0.489048	−	0.872257i	$-0.337344\pi$
$419$	8388.86			0.978097			0.489048	+	0.872257i	$0.337344\pi$
$420$	1314.14	+	1025.37i	0.152675	+	0.119126i
$421$			8187.26i			0.947797i	0.880580	+	0.473898i	$0.157154\pi$
$421$			8187.26i			0.947797i	−0.880580	+	0.473898i	$0.842846\pi$
$422$			9811.53i			1.13180i
$423$	2062.81	+	8228.56i	0.237110	+	0.945831i
$424$			11062.3i			1.26706i
$425$		−	6805.84i		−	0.776780i
$426$	−3075.01	−	2399.29i	−0.349730	−	0.272878i
$427$		−	14210.3i		−	1.61051i
$428$		−	860.312i		−	0.0971607i
$429$	−7012.17	−	5471.28i	−0.789163	−	0.615748i
$430$	−128.067			−0.0143627
$431$	−12633.2			−1.41188			−0.705941	−	0.708271i	$-0.749476\pi$
$431$	−12633.2			−1.41188			−0.705941	+	0.708271i	$0.749476\pi$
$432$	−7309.42	−	3237.55i	−0.814062	−	0.360571i
$433$	6515.64			0.723145			0.361572	−	0.932344i	$-0.382240\pi$
$433$	6515.64			0.723145			0.361572	+	0.932344i	$0.382240\pi$
$434$			14838.2i			1.64115i
$435$	8585.52	−	11003.5i	0.946308	−	1.21282i
$436$		−	831.636i		−	0.0913489i
$437$	8273.50			0.905664
$438$	−9651.93	−	7530.96i	−1.05294	−	0.821560i
$439$	−196.698			−0.0213848			−0.0106924	−	0.999943i	$-0.503404\pi$
$439$	−196.698			−0.0213848			−0.0106924	+	0.999943i	$0.503404\pi$
$440$			23913.1i			2.59094i
$441$	−1992.37	−	7947.56i	−0.215135	−	0.858175i
$442$	3901.35			0.419838
$443$	−1592.00			−0.170741			−0.0853705	−	0.996349i	$-0.527207\pi$
$443$	−1592.00			−0.170741			−0.0853705	+	0.996349i	$0.527207\pi$
$444$	400.157	+	312.224i	0.0427716	+	0.0333728i
$445$		−	3152.63i		−	0.335841i
$446$	−298.781			−0.0317213
$447$	3168.36	−	4060.67i	0.335253	−	0.429672i
$448$	14044.2			1.48109
$449$			13078.6i			1.37464i	0.726352	+	0.687322i	$0.241214\pi$
$449$			13078.6i			1.37464i	−0.726352	+	0.687322i	$0.758786\pi$
$450$	−2204.85	−	8795.16i	−0.230973	−	0.921350i
$451$		−	23569.3i		−	2.46083i
$452$	−585.052			−0.0608817
$453$	−14449.6	−	11274.4i	−1.49868	−	1.16935i
$454$	−3012.66			−0.311435
$455$	−10748.2			−1.10744
$456$	−3580.25	+	4588.56i	−0.367676	+	0.471226i
$457$			10222.0i			1.04632i	0.852236	+	0.523158i	$0.175246\pi$
$457$			10222.0i			1.04632i	−0.852236	+	0.523158i	$0.824754\pi$
$458$			304.565i			0.0310729i
$459$	−3090.16	+	6976.67i	−0.314240	+	0.709462i
$460$		−	2200.34i		−	0.223024i
$461$		−	18372.6i		−	1.85618i	−0.372354	−	0.928091i	$-0.621449\pi$
$461$		−	18372.6i		−	1.85618i	0.372354	−	0.928091i	$-0.378551\pi$
$462$	13967.5	−	17901.2i	1.40655	−	1.80268i
$463$		−	3624.32i		−	0.363794i	−0.983318	−	0.181897i	$-0.941776\pi$
$463$		−	3624.32i		−	0.363794i	0.983318	−	0.181897i	$-0.0582237\pi$
$464$		−	9677.16i		−	0.968213i
$465$	−10993.4	+	14089.5i	−1.09635	+	1.40512i
$466$	−1890.84			−0.187964
$467$	16332.5			1.61837			0.809184	−	0.587556i	$-0.199910\pi$
$467$	16332.5			1.61837			0.809184	+	0.587556i	$0.199910\pi$
$468$	−558.418	+	139.989i	−0.0551557	+	0.0138270i
$469$		−	7223.06i		−	0.711151i
$470$			13335.8i			1.30879i
$471$	5909.19	+	4610.67i	0.578091	+	0.451058i
$472$	−5311.14	+	9288.78i	−0.517935	+	0.905828i
$473$		−	193.223i		−	0.0187831i
$474$	−541.171	+	693.582i	−0.0524405	+	0.0672095i
$475$	−5936.36			−0.573429
$476$		−	1103.13i		−	0.106222i
$477$	12270.7	−	3076.13i	1.17785	−	0.295275i
$478$			13639.7i			1.30515i
$479$			2919.57i			0.278494i	0.990258	+	0.139247i	$0.0444682\pi$
$479$			2919.57i			0.278494i	−0.990258	+	0.139247i	$0.955532\pi$
$480$	2330.00	+	1818.00i	0.221562	+	0.172874i
$481$	−3272.84			−0.310247
$482$	−10284.1			−0.971845
$483$	−14173.9	+	18165.7i	−1.33527	+	1.71132i
$484$	−2209.67			−0.207520
$485$	−6841.34			−0.640514
$486$	−1733.21	+	10017.0i	−0.161769	+	0.934941i
$487$	10408.9			0.968524			0.484262	−	0.874923i	$-0.339088\pi$
$487$	10408.9			0.968524			0.484262	+	0.874923i	$0.339088\pi$
$488$		−	13195.9i		−	1.22408i
$489$	6974.29	−	8938.48i	0.644965	−	0.826609i
$490$		−	12880.3i		−	1.18750i
$491$		−	4360.92i		−	0.400826i	−0.979712	−	0.200413i	$-0.935772\pi$
$491$		−	4360.92i		−	0.400826i	0.979712	−	0.200413i	$-0.0642284\pi$
$492$	−1202.78	−	938.475i	−0.110214	−	0.0859953i
$493$	−9236.62			−0.843806
$494$		−	3402.93i		−	0.309929i
$495$	26525.3	−	6649.60i	2.40853	−	0.603793i
$496$			12391.2i			1.12173i
$497$		−	7111.35i		−	0.641827i
$498$	8074.51	−	10348.6i	0.726561	−	0.931185i
$499$	984.722			0.0883411			0.0441705	−	0.999024i	$-0.485936\pi$
$499$	984.722			0.0883411			0.0441705	+	0.999024i	$0.485936\pi$
$500$			1.70725i			0.000152701i
$501$	8689.38	+	6779.93i	0.774876	+	0.604601i
$502$		−	5747.32i		−	0.510987i
$503$	11292.6			1.00102			0.500510	−	0.865731i	$-0.333146\pi$
$503$	11292.6			1.00102			0.500510	+	0.865731i	$0.333146\pi$
$504$	−3941.32	−	15721.9i	−0.348334	−	1.38950i
$505$		−	27965.2i		−	2.46422i
$506$	−29972.8			−2.63331
$507$	−4738.97	+	6073.63i	−0.415119	+	0.532030i
$508$	291.182			0.0254313
$509$	−1779.89			−0.154994			−0.0774971	−	0.996993i	$-0.524693\pi$
$509$	−1779.89			−0.154994			−0.0774971	+	0.996993i	$0.524693\pi$
$510$	−7378.90	+	9457.04i	−0.640673	+	0.821108i
$511$		−	22321.3i		−	1.93236i
$512$	12812.1			1.10590
$513$	6085.37	+	2695.38i	0.523734	+	0.231976i
$514$		−	5982.46i		−	0.513375i
$515$	14979.3			1.28169
$516$	−9.86050	−	7.69370i	−0.000841248	−	0.000656388i
$517$	−20120.5			−1.71160
$518$		−	8355.13i		−	0.708694i
$519$	9990.74	−	12804.5i	0.844981	−	1.08296i
$520$	−9980.93			−0.841716
$521$			10556.1i			0.887662i	0.896110	+	0.443831i	$0.146381\pi$
$521$			10556.1i			0.887662i	−0.896110	+	0.443831i	$0.853619\pi$
$522$	−11936.4	+	2992.34i	−1.00085	+	0.250903i
$523$	−1374.67			−0.114934			−0.0574668	−	0.998347i	$-0.518302\pi$
$523$	−1374.67			−0.114934			−0.0574668	+	0.998347i	$0.518302\pi$
$524$	−210.448			−0.0175447
$525$	10170.0	−	13034.2i	0.845435	−	1.08354i
$526$			7245.71i			0.600623i
$527$	11827.1			0.977600
$528$	11664.0	−	14949.0i	0.961383	−	1.23214i
$529$	18248.7			1.49985
$530$	19886.7			1.62985
$531$	11780.3	+	3308.34i	0.962755	+	0.270376i
$532$	−962.198			−0.0784146
$533$	9837.41			0.799447
$534$	−1709.97	+	2191.55i	−0.138572	+	0.177599i
$535$	−17056.6			−1.37835
$536$		−	6707.41i		−	0.540515i
$537$	1627.65	−	2086.06i	0.130798	−	0.167635i
$538$	17250.4			1.38237
$539$	19433.4			1.55298
$540$	716.835	−	1618.40i	0.0571253	−	0.128972i
$541$		−	10311.3i		−	0.819443i	−0.912211	−	0.409722i	$-0.865626\pi$
$541$		−	10311.3i		−	0.819443i	0.912211	−	0.409722i	$-0.134374\pi$
$542$	6945.67			0.550446
$543$	−1453.13	+	1862.38i	−0.114843	+	0.147187i
$544$		−	1955.87i		−	0.154149i
$545$	−16488.0			−1.29591
$546$	7471.64	+	5829.78i	0.585635	+	0.456944i
$547$	24725.9			1.93273			0.966367	−	0.257168i	$-0.0827895\pi$
$547$	24725.9			1.93273			0.966367	+	0.257168i	$0.0827895\pi$
$548$			971.923i			0.0757637i
$549$	−14637.3	+	3669.42i	−1.13790	+	0.285259i
$550$	21505.9			1.66730
$551$			8056.60i			0.622909i
$552$	−13162.0	+	16868.9i	−1.01488	+	1.30070i
$553$	−1604.00			−0.123343
$554$	−19612.4			−1.50406
$555$	6190.16	−	7933.51i	0.473437	−	0.606773i
$556$	907.600			0.0692281
$557$			8023.94i			0.610387i	0.952290	+	0.305193i	$0.0987211\pi$
$557$			8023.94i			0.610387i	−0.952290	+	0.305193i	$0.901279\pi$
$558$	15284.1	−	3831.55i	1.15955	−	0.290686i
$559$	80.6479			0.00610205
$560$		−	22913.7i		−	1.72907i
$561$	−14268.4	−	11133.0i	−1.07382	−	0.837854i
$562$		−	10789.1i		−	0.809803i
$563$	−25776.6			−1.92958			−0.964791	−	0.263017i	$-0.915283\pi$
$563$	−25776.6			−1.92958			−0.964791	+	0.263017i	$0.915283\pi$
$564$	−801.152	+	1026.78i	−0.0598131	+	0.0766585i
$565$			11599.2i			0.863689i
$566$			660.464i			0.0490484i
$567$	−16343.3	+	8743.70i	−1.21050	+	0.647620i
$568$		−	6603.68i		−	0.487824i
$569$	5461.25			0.402368			0.201184	−	0.979553i	$-0.435521\pi$
$569$	5461.25			0.402368			0.201184	+	0.979553i	$0.435521\pi$
$570$	8248.86	+	6436.21i	0.606152	+	0.472953i
$571$		−	6208.29i		−	0.455007i	−0.973777	−	0.227503i	$-0.926944\pi$
$571$		−	6208.29i		−	0.455007i	0.973777	−	0.227503i	$-0.0730562\pi$
$572$		−	1365.44i		−	0.0998113i
$573$	−519.952	+	666.388i	−0.0379080	+	0.0485842i
$574$			25113.6i			1.82617i
$575$	−21823.7			−1.58280
$576$	−3626.53	−	14466.2i	−0.262335	−	1.04646i
$577$	−4213.28			−0.303988			−0.151994	−	0.988381i	$-0.548569\pi$
$577$	−4213.28			−0.303988			−0.151994	+	0.988381i	$0.548569\pi$
$578$	−5246.55			−0.377556
$579$	6837.27	−	8762.87i	0.490755	−	0.628968i
$580$	2142.65			0.153394
$581$	23932.4			1.70892
$582$	4755.76	+	3710.70i	0.338716	+	0.264285i
$583$			30004.3i			2.13148i
$584$		−	20727.8i		−	1.46870i
$585$	2775.43	+	11071.2i	0.196154	+	0.782457i
$586$			9135.46i			0.643997i
$587$	−3713.24			−0.261093			−0.130547	−	0.991442i	$-0.541673\pi$
$587$	−3713.24			−0.261093			−0.130547	+	0.991442i	$0.541673\pi$
$588$	773.794	−	991.719i	0.0542699	−	0.0695541i
$589$		−	10316.1i		−	0.721677i
$590$	16698.4	+	9547.84i	1.16519	+	0.666235i
$591$	21889.3	+	17079.2i	1.52353	+	1.18874i
$592$		−	6977.23i		−	0.484396i
$593$		−	13741.9i		−	0.951621i	−0.879548	−	0.475811i	$-0.842155\pi$
$593$		−	13741.9i		−	0.951621i	0.879548	−	0.475811i	$-0.157845\pi$
$594$	−22045.7	−	9764.67i	−1.52281	−	0.674494i
$595$	−21870.6			−1.50690
$596$	790.714			0.0543438
$597$	11682.9	−	14973.2i	0.800918	−	1.02648i
$598$		−	12510.1i		−	0.855481i
$599$			13344.3i			0.910242i	0.890430	+	0.455121i	$0.150404\pi$
$599$			13344.3i			0.910242i	−0.890430	+	0.455121i	$0.849596\pi$
$600$	9443.93	−	12103.7i	0.642578	−	0.823549i
$601$		−	11246.9i		−	0.763348i	−0.924297	−	0.381674i	$-0.875348\pi$
$601$		−	11246.9i		−	0.763348i	0.924297	−	0.381674i	$-0.124652\pi$
$602$			205.884i			0.0139389i
$603$	−7440.09	+	1865.15i	−0.502461	+	0.125962i
$604$		−	2813.70i		−	0.189549i
$605$			43808.9i			2.94394i
$606$	−15168.1	+	19440.0i	−1.01677	+	1.30313i
$607$	−25098.2			−1.67826			−0.839129	−	0.543932i	$-0.816935\pi$
$607$	−25098.2			−1.67826			−0.839129	+	0.543932i	$0.816935\pi$
$608$	−1706.00			−0.113795
$609$	−17689.5	−	13802.3i	−1.17703	−	0.918385i
$610$	−23722.2			−1.57457
$611$		−	8397.95i		−	0.556047i
$612$	−1136.27	+	284.852i	−0.0750509	+	0.0188145i
$613$			25146.8i			1.65688i	0.560074	+	0.828442i	$0.310773\pi$
$613$			25146.8i			1.65688i	−0.560074	+	0.828442i	$0.689227\pi$
$614$	−11002.9			−0.723192
$615$	−18606.2	+	23846.3i	−1.21996	+	1.56354i
$616$	38443.3			2.51449
$617$		−	10367.6i		−	0.676476i	−0.941061	−	0.338238i	$-0.890169\pi$
$617$		−	10367.6i		−	0.676476i	0.941061	−	0.338238i	$-0.109831\pi$
$618$	−10412.9	−	8124.71i	−0.677780	−	0.528841i
$619$	−19220.6			−1.24804			−0.624022	−	0.781407i	$-0.714502\pi$
$619$	−19220.6			−1.24804			−0.624022	+	0.781407i	$0.714502\pi$
$620$	−2743.57			−0.177717
$621$	22371.5	+	9908.97i	1.44563	+	0.640311i
$622$		−	3880.39i		−	0.250144i
$623$	−5068.25			−0.325931
$624$	6239.44	+	4868.35i	0.400285	+	0.312324i
$625$	−15608.1			−0.998916
$626$		−	5231.18i		−	0.333994i
$627$	−9710.71	+	12445.6i	−0.618514	+	0.792709i
$628$			1150.66i			0.0731155i
$629$	−6659.60			−0.422156
$630$	−28263.3	+	7085.31i	−1.78736	+	0.448072i
$631$	5697.06			0.359424			0.179712	−	0.983719i	$-0.442483\pi$
$631$	5697.06			0.359424			0.179712	+	0.983719i	$0.442483\pi$
$632$	−1489.49			−0.0937479
$633$	14977.3	+	11686.1i	0.940432	+	0.733777i
$634$		−	23582.7i		−	1.47727i
$635$		−	5772.98i		−	0.360777i
$636$	1531.17	+	1194.70i	0.0954637	+	0.0744859i
$637$			8111.17i			0.504515i
$638$		−	29187.0i		−	1.81117i
$639$	−7325.03	+	1836.31i	−0.453480	+	0.113683i
$640$		−	18894.9i		−	1.16701i
$641$		−	14951.6i		−	0.921296i	−0.887583	−	0.460648i	$-0.847617\pi$
$641$		−	14951.6i		−	0.921296i	0.887583	−	0.460648i	$-0.152383\pi$
$642$	11856.9	+	9251.37i	0.728899	+	0.568727i
$643$	−10278.6			−0.630401			−0.315201	−	0.949025i	$-0.602072\pi$
$643$	−10278.6			−0.630401			−0.315201	+	0.949025i	$0.602072\pi$
$644$	−3537.31			−0.216443
$645$	−152.535	+	195.494i	−0.00931173	+	0.0119342i
$646$		−	6924.32i		−	0.421724i
$647$			30130.8i			1.83086i	0.402483	+	0.915428i	$0.368147\pi$
$647$			30130.8i			1.83086i	−0.402483	+	0.915428i	$0.631853\pi$
$648$	−15176.6	+	8119.49i	−0.920050	+	0.492228i
$649$	−14405.4	+	25194.0i	−0.871283	+	1.52381i
$650$			8976.21i			0.541655i
$651$	22650.5	+	17673.2i	1.36366	+	1.06400i
$652$	1740.54			0.104547
$653$			19130.6i			1.14646i	0.819394	+	0.573231i	$0.194310\pi$
$653$			19130.6i			1.14646i	−0.819394	+	0.573231i	$0.805690\pi$
$654$	11461.6	+	8943.00i	0.685299	+	0.534708i
$655$			4172.33i			0.248896i
$656$			20972.0i			1.24820i
$657$	−22992.0	+	5763.85i	−1.36530	+	0.342267i
$658$	21438.9			1.27017
$659$	−3783.93			−0.223674			−0.111837	−	0.993727i	$-0.535673\pi$
$659$	−3783.93			−0.223674			−0.111837	+	0.993727i	$0.535673\pi$
$660$	3309.90	+	2582.56i	0.195208	+	0.152312i
$661$	14846.5			0.873619			0.436809	−	0.899554i	$-0.356108\pi$
$661$	14846.5			0.873619			0.436809	+	0.899554i	$0.356108\pi$
$662$	7977.32			0.468350
$663$	4646.73	−	5955.40i	0.272193	−	0.348852i
$664$	22223.8			1.29887
$665$			19076.5i			1.11242i
$666$	−8606.18	+	2157.48i	−0.500725	+	0.125526i
$667$			29618.3i			1.71938i
$668$			1692.04i			0.0980044i
$669$	−355.866	+	456.089i	−0.0205659	+	0.0263579i
$670$	−12057.9			−0.695280
$671$		−	35791.2i		−	2.05917i
$672$	2922.65	−	3745.77i	0.167773	−	0.215024i
$673$		−	25796.1i		−	1.47752i	−0.673971	−	0.738758i	$-0.735413\pi$
$673$		−	25796.1i		−	1.47752i	0.673971	−	0.738758i	$-0.264587\pi$
$674$			11813.8i			0.675150i
$675$	−16051.9	−	7109.83i	−0.915315	−	0.405419i
$676$	−1182.69			−0.0672898
$677$			19445.2i			1.10390i	0.833878	+	0.551949i	$0.186116\pi$
$677$			19445.2i			1.10390i	−0.833878	+	0.551949i	$0.813884\pi$
$678$	6291.36	−	8063.22i	0.356369	−	0.456735i
$679$			10998.3i			0.621614i
$680$	−20309.3			−1.14533
$681$	−3588.26	+	4598.83i	−0.201912	+	0.258778i
$682$			37372.6i			2.09835i
$683$	16732.4			0.937404			0.468702	−	0.883356i	$-0.344722\pi$
$683$	16732.4			0.937404			0.468702	+	0.883356i	$0.344722\pi$
$684$	248.461	+	991.109i	0.0138891	+	0.0554035i
$685$	19269.4			1.07481
$686$	2697.83			0.150151
$687$	464.918	+	362.754i	0.0258191	+	0.0201455i
$688$			171.930i			0.00952728i
$689$	−12523.3			−0.692451
$690$	30325.1	+	23661.3i	1.67313	+	1.30547i
$691$		−	16776.9i		−	0.923620i	−0.886979	−	0.461810i	$-0.847200\pi$
$691$		−	16776.9i		−	0.923620i	0.886979	−	0.461810i	$-0.152800\pi$
$692$	2493.35			0.136969
$693$	−10690.0	−	42642.6i	−0.585976	−	2.33746i
$694$	−12898.6			−0.705509
$695$		−	17994.1i		−	0.982092i
$696$	−16426.6	−	12816.9i	−0.894611	−	0.698024i
$697$	20017.2			1.08782
$698$			24183.7i			1.31142i
$699$	−2252.10	+	2886.36i	−0.121863	+	0.156183i
$700$	2538.07			0.137043
$701$	27374.7			1.47493			0.737466	−	0.675385i	$-0.236022\pi$
$701$	27374.7			1.47493			0.737466	+	0.675385i	$0.236022\pi$
$702$	4075.61	−	9201.52i	0.219122	−	0.494714i
$703$			5808.81i			0.311641i
$704$	35372.8			1.89370
$705$	20357.0	+	15883.6i	1.08750	+	0.848529i
$706$	30322.4			1.61643
$707$	−44957.4			−2.39151
$708$	712.101	+	1738.30i	0.0378000	+	0.0922731i
$709$	−32178.8			−1.70452			−0.852258	−	0.523122i	$-0.824767\pi$
$709$	−32178.8			−1.70452			−0.852258	+	0.523122i	$0.824767\pi$
$710$	−11871.4			−0.627503
$711$	414.187	+	1652.19i	0.0218470	+	0.0871478i
$712$	−4706.42			−0.247726
$713$		−	37924.9i		−	1.99200i
$714$	15203.4	+	11862.5i	0.796879	+	0.621768i
$715$	−27071.3			−1.41596
$716$	406.207			0.0212020
$717$	20820.9	+	16245.6i	1.08448	+	0.846170i
$718$		−	21150.1i		−	1.09933i
$719$	24046.9			1.24728			0.623641	−	0.781711i	$-0.285653\pi$
$719$	24046.9			1.24728			0.623641	+	0.781711i	$0.285653\pi$
$720$	−23602.2	+	5916.82i	−1.22167	+	0.306260i
$721$		−	24081.1i		−	1.24387i
$722$	12367.8			0.637511
$723$	−12249.0	+	15698.7i	−0.630075	+	0.807526i
$724$	−362.651			−0.0186158
$725$		−	21251.6i		−	1.08864i
$726$	23761.7	−	30453.8i	1.21471	−	1.55681i
$727$	22341.6			1.13976			0.569878	−	0.821729i	$-0.306990\pi$
$727$	22341.6			1.13976			0.569878	+	0.821729i	$0.306990\pi$
$728$			16045.6i			0.816879i
$729$	13226.6	+	14576.6i	0.671982	+	0.740567i
$730$	−37262.4			−1.88924
$731$	164.103			0.00830311
$732$	−1826.49	−	1425.12i	−0.0922253	−	0.0719592i
$733$	7788.17			0.392445			0.196223	−	0.980559i	$-0.437132\pi$
$733$	7788.17			0.392445			0.196223	+	0.980559i	$0.437132\pi$
$734$			11448.8i			0.575728i
$735$	−19661.8	−	15341.2i	−0.986718	−	0.769891i
$736$	−6271.72			−0.314102
$737$		−	18192.5i		−	0.909267i
$738$	25868.2	−	6484.88i	1.29027	−	0.323458i
$739$			11705.7i			0.582680i	0.956620	+	0.291340i	$0.0941011\pi$
$739$			11705.7i			0.582680i	−0.956620	+	0.291340i	$0.905899\pi$
$740$	1544.85			0.0767431
$741$	−5194.57	−	4053.09i	−0.257527	−	0.200936i
$742$		−	31970.3i		−	1.58176i
$743$		−	3213.88i		−	0.158689i	−0.996847	−	0.0793443i	$-0.974717\pi$
$743$		−	3213.88i		−	0.158689i	0.996847	−	0.0793443i	$-0.0252827\pi$
$744$	21033.5	+	16411.5i	1.03646	+	0.808703i
$745$		−	15676.7i		−	0.770939i
$746$	−14455.9			−0.709477
$747$	−6179.86	−	24651.5i	−0.302690	−	1.20743i
$748$		−	2778.42i		−	0.135814i
$749$			27420.5i			1.33768i
$750$	−23.5294	−	18.3589i	−0.00114556	−	0.000893832i
$751$			30502.5i			1.48209i	0.671453	+	0.741047i	$0.265671\pi$
$751$			30502.5i			1.48209i	−0.671453	+	0.741047i	$0.734329\pi$
$752$	17903.2			0.868170
$753$	−8773.27	−	6845.39i	−0.424589	−	0.331288i
$754$	12182.2			0.588393
$755$	−55784.4			−2.68901
$756$	−2601.78	−	1152.40i	−0.125166	−	0.0554397i
$757$	−12073.7			−0.579689			−0.289845	−	0.957074i	$-0.593604\pi$
$757$	−12073.7			−0.579689			−0.289845	+	0.957074i	$0.593604\pi$
$758$	6041.76			0.289507
$759$	−35699.3	+	45753.4i	−1.70725	+	2.18807i
$760$			17714.7i			0.845498i
$761$			1492.69i			0.0711040i	0.999368	+	0.0355520i	$0.0113189\pi$
$761$			1492.69i			0.0711040i	−0.999368	+	0.0355520i	$0.988681\pi$
$762$	−3131.23	+	4013.09i	−0.148861	+	0.190786i
$763$			26506.5i			1.25767i
$764$	−129.762			−0.00614480
$765$	5647.47	+	22527.8i	0.266908	+	1.06470i
$766$			19240.0i			0.907530i
$767$	−10515.5	−	6012.58i	−0.495038	−	0.283053i
$768$	3876.38	−	4968.10i	0.182131	−	0.233425i
$769$			15205.3i			0.713026i	0.934290	+	0.356513i	$0.116035\pi$
$769$			15205.3i			0.713026i	−0.934290	+	0.356513i	$0.883965\pi$
$770$		−	69109.4i		−	3.23446i
$771$	−9132.21	−	7125.45i	−0.426574	−	0.332836i
$772$	1706.35			0.0795503
$773$	19916.9			0.926727			0.463364	−	0.886168i	$-0.346642\pi$
$773$	19916.9			0.926727			0.463364	+	0.886168i	$0.346642\pi$
$774$	212.070	−	53.1636i	0.00984844	−	0.00246890i
$775$			27211.7i			1.26125i
$776$			10213.1i			0.472462i
$777$	−12754.1	−	9951.44i	−0.588868	−	0.459467i
$778$			6506.20i			0.299818i
$779$		−	17459.9i		−	0.803039i
$780$	−1077.92	+	1381.50i	−0.0494816	+	0.0634173i
$781$		−	17911.2i		−	0.820630i
$782$		−	25455.7i		−	1.16406i
$783$	−9649.19	+	21785.0i	−0.440401	+	0.994295i
$784$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.68371 q^{2} +(-3.19645 + 4.09667i) q^{3} -0.797724 q^{4} +15.8157i q^{5} +(8.57832 - 10.9943i) q^{6} +25.4256 q^{7} +23.6105 q^{8} +(-6.56545 - 26.1896i) q^{9} +O(q^{10})\)	\(q-2.68371 q^{2} +(-3.19645 + 4.09667i) q^{3} -0.797724 q^{4} +15.8157i q^{5} +(8.57832 - 10.9943i) q^{6} +25.4256 q^{7} +23.6105 q^{8} +(-6.56545 - 26.1896i) q^{9} -42.4446i q^{10} +64.0388 q^{11} +(2.54988 - 3.26801i) q^{12} +26.7287i q^{13} -68.2349 q^{14} +(-64.7916 - 50.5539i) q^{15} -56.9818 q^{16} +54.3878i q^{17} +(17.6197 + 70.2852i) q^{18} +47.4395 q^{19} -12.6165i q^{20} +(-81.2717 + 104.160i) q^{21} -171.861 q^{22} +174.401 q^{23} +(-75.4697 + 96.7245i) q^{24} -125.135 q^{25} -71.7320i q^{26} +(128.276 + 56.8171i) q^{27} -20.2826 q^{28} +169.829i q^{29} +(173.882 + 135.672i) q^{30} -217.458i q^{31} -35.9615 q^{32} +(-204.697 + 262.346i) q^{33} -145.961i q^{34} +402.123i q^{35} +(5.23742 + 20.8921i) q^{36} +122.447i q^{37} -127.314 q^{38} +(-109.499 - 85.4369i) q^{39} +373.416i q^{40} -368.046i q^{41} +(218.109 - 279.536i) q^{42} -3.01728i q^{43} -51.0853 q^{44} +(414.206 - 103.837i) q^{45} -468.041 q^{46} -314.192 q^{47} +(182.139 - 233.436i) q^{48} +303.463 q^{49} +335.826 q^{50} +(-222.809 - 173.848i) q^{51} -21.3221i q^{52} +468.533i q^{53} +(-344.256 - 152.480i) q^{54} +1012.82i q^{55} +600.312 q^{56} +(-151.638 + 194.344i) q^{57} -455.771i q^{58} +(-224.948 + 393.417i) q^{59} +(51.6858 + 40.3281i) q^{60} -558.899i q^{61} +583.593i q^{62} +(-166.931 - 665.887i) q^{63} +552.365 q^{64} -422.733 q^{65} +(549.346 - 704.060i) q^{66} -284.086i q^{67} -43.3865i q^{68} +(-557.464 + 714.464i) q^{69} -1079.18i q^{70} -279.692i q^{71} +(-155.014 - 618.349i) q^{72} -877.906i q^{73} -328.611i q^{74} +(399.988 - 512.638i) q^{75} -37.8436 q^{76} +1628.23 q^{77} +(293.863 + 229.288i) q^{78} -63.0858 q^{79} -901.206i q^{80} +(-642.790 + 343.893i) q^{81} +987.728i q^{82} +941.269 q^{83} +(64.8323 - 83.0913i) q^{84} -860.180 q^{85} +8.09748i q^{86} +(-695.733 - 542.849i) q^{87} +1511.99 q^{88} -199.336 q^{89} +(-1111.61 + 278.668i) q^{90} +679.594i q^{91} -139.124 q^{92} +(890.854 + 695.093i) q^{93} +843.199 q^{94} +750.288i q^{95} +(114.949 - 147.322i) q^{96} +432.568i q^{97} -814.404 q^{98} +(-420.444 - 1677.15i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + O(q^{10}) \)	\( 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + 28q^{12} + 114q^{15} + 484q^{16} - 184q^{19} - 758q^{21} - 60q^{22} + 36q^{25} + 742q^{27} - 4q^{28} - 888q^{36} + 1402q^{45} - 660q^{46} - 488q^{48} - 924q^{49} - 1772q^{51} - 630q^{57} - 1880q^{60} - 212q^{63} + 7648q^{64} + 1316q^{66} - 1556q^{75} - 5680q^{76} + 3224q^{78} - 1504q^{79} - 276q^{81} + 1228q^{84} - 848q^{85} + 3598q^{87} + 5760q^{88} + 888q^{94} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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